Divide using synthetic division $$ ( 2x^{4}-7x^{3}-17x^{2}+59x-24 ) \div ( x+24 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-24&2&-7&-17&59&-24\\& & -48& 1320& -31272& \color{black}{749112} \\ \hline &\color{blue}{2}&\color{blue}{-55}&\color{blue}{1303}&\color{blue}{-31213}&\color{orangered}{749088} \end{array} $$The solution is:
$$ \dfrac{ 2x^{4}-7x^{3}-17x^{2}+59x-24 }{ x+24 } = \color{blue}{2x^{3}-55x^{2}+1303x-31213} ~+~ \dfrac{ \color{red}{ 749088 } }{ x+24 } $$Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 24 = 0 $ ( $ x = \color{blue}{ -24 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{-24}&2&-7&-17&59&-24\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-24&\color{orangered}{ 2 }&-7&-17&59&-24\\& & & & & \\ \hline &\color{orangered}{2}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -24 } \cdot \color{blue}{ 2 } = \color{blue}{ -48 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-24}&2&-7&-17&59&-24\\& & \color{blue}{-48} & & & \\ \hline &\color{blue}{2}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ \left( -48 \right) } = \color{orangered}{ -55 } $
$$ \begin{array}{c|rrrrr}-24&2&\color{orangered}{ -7 }&-17&59&-24\\& & \color{orangered}{-48} & & & \\ \hline &2&\color{orangered}{-55}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -24 } \cdot \color{blue}{ \left( -55 \right) } = \color{blue}{ 1320 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-24}&2&-7&-17&59&-24\\& & -48& \color{blue}{1320} & & \\ \hline &2&\color{blue}{-55}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -17 } + \color{orangered}{ 1320 } = \color{orangered}{ 1303 } $
$$ \begin{array}{c|rrrrr}-24&2&-7&\color{orangered}{ -17 }&59&-24\\& & -48& \color{orangered}{1320} & & \\ \hline &2&-55&\color{orangered}{1303}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -24 } \cdot \color{blue}{ 1303 } = \color{blue}{ -31272 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-24}&2&-7&-17&59&-24\\& & -48& 1320& \color{blue}{-31272} & \\ \hline &2&-55&\color{blue}{1303}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 59 } + \color{orangered}{ \left( -31272 \right) } = \color{orangered}{ -31213 } $
$$ \begin{array}{c|rrrrr}-24&2&-7&-17&\color{orangered}{ 59 }&-24\\& & -48& 1320& \color{orangered}{-31272} & \\ \hline &2&-55&1303&\color{orangered}{-31213}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -24 } \cdot \color{blue}{ \left( -31213 \right) } = \color{blue}{ 749112 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-24}&2&-7&-17&59&-24\\& & -48& 1320& -31272& \color{blue}{749112} \\ \hline &2&-55&1303&\color{blue}{-31213}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -24 } + \color{orangered}{ 749112 } = \color{orangered}{ 749088 } $
$$ \begin{array}{c|rrrrr}-24&2&-7&-17&59&\color{orangered}{ -24 }\\& & -48& 1320& -31272& \color{orangered}{749112} \\ \hline &\color{blue}{2}&\color{blue}{-55}&\color{blue}{1303}&\color{blue}{-31213}&\color{orangered}{749088} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{3}-55x^{2}+1303x-31213 } $ with a remainder of $ \color{red}{ 749088 } $.